Chapter 3 - Rings of differential operators

Summary

In this chapter we show that the Weyl algebras are members of the family of rings of differential operators. These rings come up in many areas of mathematics: representation theory of Lie algebras, singularity theory and differential equations are some of them.

DEFINITIONS.

Let R be a commutative K-algebra. The ring of differential operators of R is defined, inductively, as a subring of End_K(R). As in the case of the Weyl algebra, we will identify an element a ∈ R with the operator of End_K(R) defined by the rule r ↦ ar, for every r ∈ R.

We now define, inductively, the order of an operator. An operator P ∈ End_K(R) has order zero if [a, P] = 0, for every a ∈ R. Suppose we have defined operators of order < n. An operator P ∈ End_K(R) has order n if it does not have order less than n and [a, P] has order less than n for every a ∈ R. Let Dⁿ(R) denote the set of all operators of End_K(R) of order ≤ n. It is easy to check, from the definitions, that Dⁿ(R) is a K-vector space.

We may characterize the operators of order ≤ 1 in terms of well-known concepts. A derivation of the K-algebra R is a linear operator D of which satisfies Leibniz's rule: D(ab) = aD(b) + bD(a) for every a,b ∈ R. Let Der_K(R) denote the K-vector space of all derivations of R. Of course Der_K(R) ⊆ End_K(R).